In this work, we study a nonparametric regression model with centered errors and with response variables that are missing at random (MAR). Our main goal is to find an estimator for the expectation of a function of the observations which is efficient in the Hájek-Le Cam-sense. In some similar models such as a linear or a parametric regression model, other authors constructed efficient estimators for the same expectation value. The problem considered here has not yet been dealed with even in the case of non-missing response variables. Therefore, in a first step, we have to prove local asymptotic normality (LAN) of the model. As the canonical gradient of a suitable functional is essential for efficiency, it is calculated next. Then an estimator is constructed via full imputation of response variables with nonparametric estimators of conditional expectations. To do so the regression function is estimated by a truncated version of the Nadaraya-Watson estimator. In order to examine whether the estimator constructed above is efficient, asymptotic linearity of the estimator and its influence function are derived and compared with the canonical gradient. It turns out, however, that the resulting estimator is not efficient. This differs from the result in the linear and the parametric regression model. Nevertheless the determined influence function matches the desired one quite well. After adding a correction term we finally obtain an efficient estimator. The last step may also be regarded as second part of this work. It contains arguments which had not been used in the case of missing data before.